Chapter 1: Mathematics, Lesson 1: Numbers and Operations
The Mathematics category is Content Category I on your Praxis 5155 exam, worth 50% of your total score (40 of the 80 questions). Numbers and Operations is the foundation that every other math lesson in this course builds on: fractions, algebra, geometry, and statistics all depend on the skills you review here. Each section below teaches one testable skill, moves straight to an example, and shows you the answer and the reasoning behind it.
I. Mathematics: 50% (this chapter, 40 questions)
II. Science: 50% (Chapter 2, 40 questions)
Learning Outcomes
After studying this lesson, you will be able to:
- Add, subtract, multiply, and divide rational numbers, and apply the order of operations.
- Read, write, and compare numbers using place value, and identify the commutative, associative, distributive, and identity properties.
- Order positive and negative integers, and work with positive exponents.
- Convert among fractions, decimals, and percents, and represent them using different visual models.
- Use ratios, unit rates, scale factors, and percent formulas to solve applied problems.
- Apply number theory concepts, including primes, composites, factors, multiples, prime factorization, GCF, and LCM.
- Use estimation and rounding to judge whether a computed answer is reasonable.
(1) Basic Operations and Properties of Rational Numbers
A rational number is any number that can be written as a fraction of two integers, including whole numbers, fractions, and terminating or repeating decimals. On the exam, you must add, subtract, multiply, and divide these numbers accurately, including when they are negative.
(a) Addition, Subtraction, Multiplication, and Division of Rational Numbers
Addition example: Add −3/4 + 1/2.
Rewrite 1/2 with a denominator of 4: 1/2 = 2/4.
−3/4 + 2/4 = −1/4
Answer: −1/4
Why: You cannot add fractions with different denominators directly. Once both fractions share the denominator 4, you add the numerators (−3 + 2 = −1) and keep the denominator.
Subtraction example: Subtract −2.5 − (−6.75).
Subtracting a negative number is the same as adding its positive opposite: −2.5 − (−6.75) = −2.5 + 6.75
−2.5 + 6.75 = 4.25
Answer: 4.25
Why: Two negative signs next to each other (subtracting a negative) always turn into a positive sign. Once the expression becomes −2.5 + 6.75, you find the difference between 6.75 and 2.5 and keep the sign of the larger absolute value, which is positive.
Multiplication example: Multiply −2/3 × 9/4.
Multiply numerators and denominators: (−2 × 9) / (3 × 4) = −18/12
Simplify by dividing the numerator and denominator by 6: −18/12 = −3/2
Answer: −3/2 (or −1 1/2)
Why: A positive number times a negative number always gives a negative product. Multiply straight across, then simplify the resulting fraction to lowest terms.
Division example: Divide −5/6 ÷ 2/3.
Dividing by a fraction means multiplying by its reciprocal: −5/6 ÷ 2/3 = −5/6 × 3/2
(−5 × 3) / (6 × 2) = −15/12, which simplifies to −5/4
Answer: −5/4 (or −1 1/4)
Why: Division of fractions is always converted into multiplication by flipping the second fraction (its reciprocal). The sign rule is the same as multiplication: a negative divided by a positive gives a negative answer.
⚠ COMMON TRAP: Students often forget to find a common denominator before adding or subtracting fractions, or forget to flip the second fraction before multiplying in a division problem. Always check: same denominator before adding/subtracting; reciprocal of the second fraction before dividing.
(b) Order of Operations
When an expression has more than one operation, you must complete them in a fixed order, remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).
Parentheses
Exponents
left to right
left to right
Example: Evaluate 6 + 3 × (8 − 2)2 ÷ 4 − 5.
| Step | Operation | Result |
|---|---|---|
| Parentheses | (8 − 2) | 6 + 3 × 62 ÷ 4 − 5 |
| Exponent | 62 | 6 + 3 × 36 ÷ 4 − 5 |
| Multiply | 3 × 36 | 6 + 108 ÷ 4 − 5 |
| Divide | 108 ÷ 4 | 6 + 27 − 5 |
| Add/Subtract, left to right | 6 + 27, then − 5 | 33 − 5 |
Answer: 28
Why: Skipping the order of operations and working strictly left to right would give a different, incorrect result. Multiplication and division are equal in priority and completed left to right; the same is true for addition and subtraction.
★ TEST READY TIP: On the Praxis 5155 exam, multiplication and division share one level of priority, and addition and subtraction share another. Within each level you work strictly left to right, so in 20 − 8 + 3 you subtract first (12) and then add (15), not the reverse. Reading M before D or A before S as a strict order is the single most common order-of-operations error.
(c) Place Value: Reading and Writing Numbers
Place value tells you what a digit is worth based on its position in a number. Study the color-coded table below, then look at how the number 4,208,617.359 is labeled digit by digit.
| Millions | Hundred Thousands | Ten Thousands | Thousands | Hundreds | Tens | Ones | . | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|---|---|---|---|---|---|
| 4 | 2 | 0 | 8 | 6 | 1 | 7 | . | 3 | 5 | 9 |
Reading this left to right: 4 millions, 2 hundred thousands, 0 ten thousands, 8 thousands, 6 hundreds, 1 ten, 7 ones, 3 tenths, 5 hundredths, 9 thousandths, so the number is read aloud as "four million, two hundred eight thousand, six hundred seventeen, and three hundred fifty-nine thousandths."
Standard form is the number written the normal way (4,208,617.359). Expanded form breaks it into the sum of each digit's true value. Here are 5 examples:
| Standard Form | Expanded Form |
|---|---|
| 3,942,105 | 3,000,000 + 900,000 + 40,000 + 2,000 + 100 + 5 |
| 56.24 | 50 + 6 + 0.2 + 0.04 |
| 771,144 | 700,000 + 70,000 + 1,000 + 100 + 40 + 4 |
| 0.638 | 0.6 + 0.03 + 0.008 |
| 12,050.07 | 10,000 + 2,000 + 50 + 0.07 |
(d) Properties of Operations
| Property | Rule | Labeled Example |
|---|---|---|
| Commutative | Changing the order of the numbers does not change the result (addition and multiplication only) | 7 + 12 = 12 + 7 (both equal 19) |
| Associative | Changing which numbers are grouped first does not change the result (addition and multiplication only) | (2 × 5) × 6 = 2 × (5 × 6) (both equal 60) |
| Distributive | Multiplying a number by a sum gives the same result as multiplying it by each addend, then adding | 4 × (9 + 3) = (4 × 9) + (4 × 3) (both equal 48) |
| Identity | Adding 0 or multiplying by 1 leaves a number unchanged | 15 + 0 = 15 (additive identity) • 8 × 1 = 8 (multiplicative identity) |
(e) Ordering Positive and Negative Integers
On a number line, values increase as you move right and decrease as you move left. The position of a number on the line, not how "big" its digits look, tells you which of two numbers is larger.
How to know which is bigger: compare the two numbers' positions on the line. Since −2 sits to the right of −5, −2 > −5. Since 3 sits to the right of −5 and −2, 3 is greater than both. For two negative numbers, the one closer to zero (the smaller absolute value) is always the greater number: −3 > −8, because −3 is closer to zero.
(f) Positive Exponents
An exponent tells you how many times to multiply the base number by itself.
| Expression | Expanded | Answer |
|---|---|---|
| 53 | 5 × 5 × 5 | 125 |
| 26 | 2 × 2 × 2 × 2 × 2 × 2 | 64 |
| 122 | 12 × 12 | 144 |
Why: Writing out the expanded multiplication (as shown in the middle column) shows exactly why the exponent rule works: the small raised number is only a count of how many bases are being multiplied, never a multiplier applied to the base itself.
(2) Relationships Among Fractions, Decimals, and Percents
Fractions, decimals, and percents are three different ways to express the same value. You need to move fluidly between all three forms, including values greater than 1, negative values, and fractions that convert to repeating decimals.
| Fraction | Decimal | Percent |
|---|---|---|
| 7/16 | 0.4375 | 43.75% |
| 4/7 | 0.571428… | 57.14% (repeating) |
| 11/8 | 1.375 | 137.5% |
| −9/20 | −0.45 | −45% |
| 17/40 | 0.425 | 42.5% |
| 5/6 | 0.83… | 83.3% (repeating) |
Notice the value greater than 1 (11/8, worth more than 100%) and the negative value (−9/20). A repeating decimal is marked with a bar over the digit or digits that repeat forever; on the exam, you can also write "..." after enough repeating digits to show the pattern, or round to a stated number of places if the question asks you to.
(a) Simplifying Fractions to Lowest Terms
There are three types of fractions you must recognize, and every one of them can be simplified.
| Type | Definition | Simplify Example |
|---|---|---|
| Proper fraction | Numerator is smaller than the denominator | 84/126 = (84 ÷ 42)/(126 ÷ 42) = 2/3 |
| Improper fraction | Numerator is equal to or larger than the denominator | 132/48 = (132 ÷ 12)/(48 ÷ 12) = 11/4 |
| Mixed number | A whole number joined with a proper fraction | 9 108/144 → simplify 108/144 (divide by the GCF, 36) to 3/4 = 9 3/4 |
To simplify any fraction, divide the numerator and denominator by their greatest common factor (GCF). The notation (84 ÷ 42)/(126 ÷ 42) means you divide the numerator and the denominator each by the GCF, which shrinks the fraction to lowest terms without changing its value. The GCF is not always obvious. For 84/126, list factors or use prime factorization (84 = 22 × 3 × 7, 126 = 2 × 32 × 7) to find the shared factors 2, 3, and 7, which multiply to a GCF of 42. An improper fraction (like 11/4) can also be rewritten as a mixed number (2 3/4) by dividing the numerator by the denominator and keeping the remainder as the new numerator.
(b) Equivalent Fractions
Equivalent fractions represent the same value even though they use different numbers. Multiply (or divide) the numerator and denominator by the same number to create one. The bars below illustrate the concept with a simple case (one half); the numbered examples that follow show the level of numbers you can expect on the exam.
The shaded bars below show the same amount, one half, split into different numbers of equal pieces:
Missing-value example: Find the missing numerator: 5/12 = ?/60.
First find what the denominator was multiplied by: 60 ÷ 12 = 5.
Multiply the numerator by that same number: 5 × 5 = 25.
Answer: 5/12 = 25/60
Why: Whatever number you multiply the denominator by, you must multiply the numerator by the same number, or the value of the fraction changes.
Extended chain example: Generate two equivalent fractions for 7/18.
7/18 × 2/2 = 14/36
7/18 × 5/5 = 35/90
Answer: 7/18 = 14/36 = 35/90
(c) Converting Among Fractions, Decimals, and Percents
| Conversion | Example | Method |
|---|---|---|
| Fraction → Decimal | 5/8 = 0.625 | Divide the numerator by the denominator (5 ÷ 8) |
| Fraction → Percent | 7/20 = 35% | Divide to get 0.35, then multiply by 100 |
| Decimal → Fraction | 0.45 = 9/20 | Write as 45/100, then simplify by dividing by 5 |
| Decimal → Percent | 0.6 = 60% | Multiply by 100 (move the decimal point 2 places right) |
| Percent → Decimal | 82% = 0.82 | Divide by 100 (move the decimal point 2 places left) |
| Percent → Fraction | 15% = 3/20 | Write as 15/100, then simplify by dividing by 5 |
(d) Representing Fractions, Decimals, and Percents With Models
Your exam may show a value as a picture instead of a number. You need to be able to read each of these models on sight.
Shaded area model: A rectangle split into 5 equal columns with 3 shaded represents 3/5.
3 of 5 equal parts are shaded: 3/5 = 0.6 = 60%.
10-by-10 percent grid: Each small square is worth 1% of the whole grid (100 squares total).
62 of the 100 squares are shaded: 62% = 0.62 = 62/100 = 31/50.
(e) Comparing Positive Rational Numbers
| Scenario | Example | How to Compare |
|---|---|---|
| Same denominator | 5/9 vs. 7/9 | Compare numerators directly: 7/9 > 5/9 |
| Different denominators | 3/4 vs. 5/7 | Cross-multiply: 3 × 7 = 21, 5 × 4 = 20, so 3/4 > 5/7 |
| Fraction vs. decimal | 5/8 vs. 0.6 | Convert the fraction: 5/8 = 0.625, so 5/8 > 0.6 |
| Decimal vs. decimal | 0.45 vs. 0.5 | Line up place value: 0.45 vs. 0.50, so 0.5 > 0.45 |
(3) Ratios and Proportional Relationships
(a) The Concept of a Ratio
A ratio compares two quantities. A classroom of 28 students has 12 boys and 16 girls.
Ratio of boys to girls: 12:16, which simplifies (divide both terms by 4) to 3:4.
Why: A ratio just compares two numbers, so it can always be simplified the same way a fraction is, by dividing both terms by their greatest common factor.
(b) Using a Proportion to Solve a Problem
Unit rate example: A car travels 210 miles on 7 gallons of gas. What is the unit rate in miles per gallon?
Set up the proportion: 210 miles / 7 gallons = x miles / 1 gallon
210 ÷ 7 = 30
Answer: 30 miles per gallon
Scale factor example: A blueprint uses a scale where 1 inch represents 4 feet of an actual wall. A wall measures 5.5 inches on the blueprint. How long is the actual wall?
Set up the proportion: 1 inch / 4 feet = 5.5 inches / x feet
5.5 × 4 = 22
Answer: 22 feet
(c) Solving Percent Problems
| Type | Problem | Solution |
|---|---|---|
| Discount | A $84 jacket is 25% off. | 84 × 0.25 = 21 discount; 84 − 21 = $63 |
| Sales tax | A $56 meal has 8% tax. | 56 × 0.08 = 4.48 tax; 56 + 4.48 = $60.48 |
| Gratuity (tip) | A $72 bill gets an 18% tip. | 72 × 0.18 = 12.96 tip; 72 + 12.96 = $84.96 |
| Simple interest | $2,000 principal at 5% annual rate for 3 years. | I = Prt = 2,000 × 0.05 × 3 = $300 |
Simple interest formula: Simple interest is I = Prt, where P is the principal (starting amount), r is the annual interest rate written as a decimal, and t is the time in years.
(4) Basic Concepts of Number Theory
Number theory is the branch of mathematics that studies whole numbers and how they divide into one another.
(a) Prime vs. Composite Numbers
PRIME
- Exactly 2 factors: 1 and itself
- 2, 3, 5, 7, 11, 13, 17, 19…
- 2 is the only even prime number
- 1 is neither prime nor composite
COMPOSITE
- More than 2 factors
- 4, 6, 8, 9, 10, 12, 14, 15…
- Every composite number breaks down into a product of prime numbers
- Any even number greater than 2 is composite
(b) Odd vs. Even Numbers
EVEN
- Divisible by 2 with no remainder
- Ends in 0, 2, 4, 6, or 8
- Even + Even = Even
ODD
- Not evenly divisible by 2
- Ends in 1, 3, 5, 7, or 9
- Odd + Odd = Even · Odd + Even = Odd
(c) Solving Problems With Factors and Multiples
Factors example: Find all factors of 36.
List every number that divides evenly into 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
Common factors and GCF example: Find the greatest common factor (GCF) of 24 and 36.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Common factors: 1, 2, 3, 4, 6, 12 → GCF = 12
Prime factorization example: Write 60 as a product of prime numbers.
60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5, so 60 = 22 × 3 × 5
Multiples example: List the first 5 multiples of 8.
8, 16, 24, 32, 40 (each is 8 times a counting number)
Common multiples and LCM example: Two classroom timers beep, one every 6 minutes and the other every 8 minutes, and both beep together at the start. When will they beep together again?
Multiples of 6: 6, 12, 18, 24, 30…
Multiples of 8: 8, 16, 24, 32…
Answer: 24 minutes (the least common multiple, LCM)
★ TEST READY TIP: Word problems that ask you to split a group into equal sub-groups with nothing left over are factor problems. Word problems where two repeating events line up again are multiple (LCM) problems.
(5) Estimation and Reasonableness
Estimation lets you check whether a computed answer makes sense. Before you check out at the grocery store, rounding each item to the nearest dollar and adding the rounded amounts gives you a quick total; if the register then rings up an amount far from your estimate, you know to recheck for an error.
(a) Estimation and Rounding
To round a number to a given place, look at the digit one place to the right of that place. If it is 5 or more, round up; if it is 4 or less, round down.
Example 1: Round 3,847 to the nearest hundred.
3, 8 4 7 → the decision digit (4) is less than 5, so round down → 3,800
Example 2: Round 6.278 to the nearest hundredth.
6.2 7 8 → the decision digit (8) is 5 or more, so round up → 6.28
Example 3: Round 149,500 to the nearest thousand.
149, 0500 → the decision digit (5) rounds up → 150,000
Blue = the place you are rounding to. Gold = the decision digit that tells you whether to round up or down.
(b) When Estimation and Rounding Are (and Are Not) Appropriate
APPROPRIATE
- Estimating your total at the grocery store before reaching the register
- Estimating how long a road trip will take before you leave
- Checking whether a calculated answer is "in the right ballpark"
NOT APPROPRIATE
- Calculating an exact medication dosage
- Reporting exact totals on a financial or tax document
- Cutting a board to a precise construction measurement
Quick Reference Card: Numbers and Operations
- Order of operations: Parentheses → Exponents → Multiply/Divide (left to right) → Add/Subtract (left to right)
- Properties: commutative (order changes) · associative (grouping changes) · distributive (a(b + c) = ab + ac) · identity (+0 or ×1 leaves a number unchanged)
- On a number line, values increase to the right; for two negatives, the one closer to zero is greater
- Fraction types: proper (numerator < denominator) ≠ improper (numerator ≥ denominator) ≠ mixed number (whole number + proper fraction)
- Simplify a fraction by dividing the numerator and denominator by their GCF (for example, 84/126 = (84 ÷ 42)/(126 ÷ 42) = 2/3)
- Six conversions: fraction ↔ decimal (divide) · decimal ↔ percent (×100 or ÷100) · percent ↔ fraction (over 100, then simplify)
- Unit rate = ratio reduced to a denominator of 1; simple interest formula: I = Prt
- Prime = exactly 2 factors ≠ composite = more than 2 factors; 1 is neither; equal-group word problems use factors/GCF, repeating-event word problems use multiples/LCM
- To round: look one place to the right of the target place; 5 or more rounds up, 4 or less rounds down